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October, 1974 $R$-Theory for Markov Chains on a General State Space II: $r$-Subinvariant Measures for $r$-Transient Chains
Richard L. Tweedie
Ann. Probab. 2(5): 865-878 (October, 1974). DOI: 10.1214/aop/1176996553

Abstract

This paper is a sequel to a previous paper of similar title. The structure of $r$-subinvariant measures for a Markov chain $\{X_n\}$ on a general state space $(\mathscr{X}, \mathscr{F})$ is investigated in the $r$-transient case, and a Martin boundary representation is found. Under certain continuity assumptions on the transition law of $\{X_n\}$ the elements of the Martin boundary are identified when $\mathscr{F}$ is countably generated, and a necessary and sufficient condition for an $r$-invariant measure for $\{X_n\}$ to exist is found. This generalizes the Harris-Veech conditions for countable $\mathscr{X}$.

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Richard L. Tweedie. "$R$-Theory for Markov Chains on a General State Space II: $r$-Subinvariant Measures for $r$-Transient Chains." Ann. Probab. 2 (5) 865 - 878, October, 1974. https://doi.org/10.1214/aop/1176996553

Information

Published: October, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0296.60041
MathSciNet: MR368152
Digital Object Identifier: 10.1214/aop/1176996553

Subjects:
Primary: 60J05
Secondary: 45C05‎ , 45N05 , 60J45 , 60J50

Keywords: $R$-theory , boundary theory , integral equations , Invariant measures , Markov chains , potential theory , Stationary measures , subinvariant measures

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 5 • October, 1974
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